Quantitative measure equivalence between amenable groups

Delabie, Thiebout; Koivisto, Juhani; Le Maître, François; Tessera, Romain

Annales Henri Lebesgue, Tome 5 (2022), pp. 1417-1487.

Nous initions une étude quantitative de l’équivalence mesurée entre groupes de type fini. Ce faisant nous élargissons le contexte devenu classique de l’équivalence mesurée ${\mathrm{L}}^p$. Dans cet article, nous nous concentrons sur le cas des groupes moyennables, classe pour laquelle nous démontrons un théorème de rigidité ainsi que plusieurs résultats de flexibilité.

Notre résultat de rigidité stipule que le profil isopérimétrique possède une propriété de monotonicité très générale, impliquant en particulier son invariance sous équivalence mesurée intégrable. Ceci fournit des «  bornes inférieures » explicites sur le degré d’intégrabilité des couplages mesurés possibles entre deux groupes moyennables. Ce résultat a également une application inattendue en géométrie des groupes : le profil isopérimétrique est monotone sous plongement grossier entre groupes moyennables. Entre autres applications, cela entraine l’existence d’un continuum de groupes $3$-résolubles n’admettant aucun plongement grossier les uns dans les autres.

Nos résultats de flexibilité consistent à exhiber des équivalences orbitales explicites aux propriétés d’intégrabilité prescrites entre certains groupes moyennables. Nous introduisons pour cela un nouvel outil  : les suites de pavages de Følner. Nous en déduisons dans de nombreux cas que l’obstruction quantitative obtenue à l’aide du profil isopérimétrique est optimale à un facteur d’erreur logarithmique près. Nous obtenons en outre que deux invariant importants de quasi-isométrie ne sont pas préservés sous équivalence orbitale intégrable  : la dimension asymptotique, et le fait d’être de présentation finie.

We initiate a quantitative study of measure equivalence (and orbit equivalence) between finitely generated groups, which extends the classical setting of ${\mathrm{L}}^p$ measure equivalence. In this paper, our main focus will be on amenable groups, for which we prove both rigidity and flexibility results.

On the rigidity side, we prove a general monotonicity property satisfied by the isoperimetric profile, which implies in particular its invariance under ${\mathrm{L}}^1$ measure equivalence. This yields explicit “lower bounds” on how integrable a measure coupling between two amenable groups can be. This result also has an unexpected application to geometric group theory: the isoperimetric profile turns out to be monotonous under coarse embedding between amenable groups. This has various applications, among which the existence of an uncountable family of $3$-solvable groups which pairwise do not coarsely embed into one another.

On the flexibility side, we construct explicit orbit equivalences between amenable groups with prescribed integrability conditions. Our main tool is a new notion of Følner tiling sequences. We show in a number of instances that the bounds derived from the isoperimetric profile are sharp up to a logarithmic factor. We also deduce from this study that two important quasi-isometry invariants are not preserved under ${\mathrm{L}}^1$ orbit equivalence: the asymptotic dimension and finite presentability.

Mots clés : Measure equivalence, Orbit equivalence, Amenable groups

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